Question

If the $p^{th}$ term is $q$ and $q^{th}$ term is $p$ of an A.P., then find the first term and common difference.

Answer 1

Let first term be 'a' and common difference be d .

$p^{th}term=q$

$q^{th}term=p.$

$p^{th}term=a+(p-1)d=q----\left(1\right)$

$q^{th}trem=a+(q-1)d=p----\left(2\right)$

(ii)--(i)

a+(q-1)d=p

a+(p-1)d=q

___________

(q-1)d-(p-1)d=p-q

$\Rightarrow d[q-1-p+1]=p-q.$

$\Rightarrow d[q-p]=p-q$

$\Rightarrow d=\frac{(p-q)}{p-q}$

$\therefore d=-1$

sub. d=-1 in (1)

a=q-(p-1)d=q-(p-1)(-1)

=q+p-1

Hence,

first term a= q+p-1

common difference d=-1